No abstract

This article deals with the allocation of objects where each agent receives a single item. Starting from an initial endowment, the agents can be better off by exchanging their objects. However, not all trades are likely because some participants are unable to communicate. By considering that the agents are embedded in a social network, we propose to study the possible allocations emerging from a sequence of simple swaps between pairs of neighbors in the network. This model raises natural questions regarding (i) the reachability of a given full allocation, (ii) the ability of an agent to obtain a given object, and (iii) the search of Paretoefficient allocations. We investigate the complexity of these problems by providing, according to the structure of the social network, polynomial and NP-complete cases.

We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs). Finally, dealing with hypergraphs, we study the complexity and the approximability of two natural generalizations.